Collineation groups of the smallest Bol 3-nets

Abstract

Some associativity properties of a loop can be interpreted as certain closure configuration of the corresponding 3-net. It was known that the smallest non-associative loops with the so called left Bol property have order 8. In this paper, we determine the direction preserving collineation groups of the 3-nets belonging to these smallest Bol loops. For that, we prove some new results concerning collineations of 3-nets and autotopisms of loops.